This is not grammar, but about received British usage.
The topic is negating the word “different” to indicate similarity.
A is not too different from B.
That sentence clearly indicates that A and B are different. We are discussing the degree of difference between two distinguishable things.
A is not different from B
is ambiguous. It may mean that A is B; that two different names refer to the same thing. Or it may mean that two distinct things are similar in all relevant respects. The source is saying to avoid a locution that has an ambiguous meaning. Notice that this comes up in the context of the phrase “no different,” which is an idiom that unambiguously means that what are being compared are distinct but similar in all relevant respects.
Jack is no different than any other child his age
does not mean that every child has the same eye color, the same hair color, the same sex, etc. It implies some attribute that is shared by all children of that age.